# Thread: How do I show that this series diverges?

1. ## How do I show that this series diverges?

Again, I am having trouble proving what I keep getting right by using by subconscious logic.

For the problem:
a_1 = 2, a_(n+1) = (5n+1)/(4n+3) * a_n ==> Does this converge (absolutely or conditionally) or diverge?

What I did was:

the limit of (5n+1)/(4n+3) as n->inf = 5/4 therefore you will be adding increaing numbers and that means the sum will be infinite. Also, to try to prove this, I wrote that when n>=2, 5n+1 >= 4n+3 to show that I am always multiplying the previous number by something equal to or larger than 1 since the sum starts at 2 due to a_1 being predetermined.

If someone could please help me get my thoughts into mathematical writing then it would be greatly appreciated!

2. Originally Posted by s3a
Again, I am having trouble proving what I keep getting right by using by subconscious logic.

For the problem:
a_1 = 2, a_(n+1) = (5n+1)/(4n+3) * a_n ==> Does this converge (absolutely or conditionally) or diverge?

What I did was:

the limit of (5n+1)/(4n+3) as n->inf = 5/4 therefore you will be adding increaing numbers and that means the sum will be infinite. Also, to try to prove this, I wrote that when n>=2, 5n+1 >= 4n+3 to show that I am always multiplying the previous number by something equal to or larger than 1 since the sum starts at 2 due to a_1 being predetermined.

If someone could please help me get my thoughts into mathematical writing then it would be greatly appreciated!
$\displaystyle \frac{a_{n+1}}{a_n} = \frac{5n+1}{4n+3}\xrightarrow [n\to\infty]{}\frac{5}{4}>1\Longrightarrow$ the series diverges.